Two Different Ways a Probability Answer Can Go Wrong
A probability answer depends on two things: the model used to represent the chance process and the calculation made from that model. These are separate sources of error. A model may fit the process, but an arithmetic mistake can produce the wrong probability. Or the arithmetic may be flawless while the model’s assumptions fail to describe the process.
In What Makes a Probability Model Appropriate and Assumptions Behind a Probability Model, you learned to ask whether a model represents the outcomes and chance mechanism. Here, we make that check distinct from verifying the numerical work. A calculator can evaluate the expression entered correctly; it cannot tell you whether the expression represents the situation.
If the model is appropriate but the calculation is wrong, keep the model and correct the calculation. If the model is inappropriate, repeating the same calculation more carefully will not fix the problem: revise the model or use a method that matches the process. When both are in doubt, investigate the model first and then check the arithmetic.
A Two-Part Audit
A useful habit is to leave a visible trail from the situation to the conclusion. First identify what the random variable counts and what conditions the model relies on. Then check that the requested event is translated correctly, the formula or technology inputs match that event, and the arithmetic has been evaluated and reported accurately.
Identify the outcomes and the chance process. Check whether the model’s probabilities and assumptions—such as equal likelihood, a fixed success probability, or independence—are reasonable in context.
Define the random variable, translate the requested event into values of that variable, and choose a calculation that corresponds to those values.
Substitute carefully, evaluate the expression, and check the result with a second method when practical—for example, by recomputing a product or using a calculator to verify a hand calculation.
Interpret the probability under the stated model and in context. Do not claim more than the model and calculation support.
This order matters. A numerical answer can be exactly right for a model that does not fit. Conversely, a suitable model does not guarantee that the event was set up or calculated correctly. The examples below isolate these possibilities.
Worked Example: A Decimal-Place Error Changes the Conclusion
A quality-control process produces items with a stable 0.25 probability of being defective. For this example, suppose eight items are selected in a way that makes their defect statuses independent, with the same probability of defect for each item. Let \(X\) be the number of defective items among the eight. Find the probability that exactly two are defective, and check a student’s reported answer of \(0.0311\).
State and model check. The outcomes are defective or not defective for each item, and \(X\) counts defects in eight trials. The process description supports a fixed success probability \(p=0.25\), and the items are specified to be independent. Thus the binomial model \(X\sim B(8,0.25)\) is appropriate for this calculation. These are the BINS checks reviewed in Common Errors with Binomial Distributions.
Plan and do. “Exactly two” means \(X=2\). Use the binomial probability formula, with \(n=8\), \(p=0.25\), and \(1-p=0.75\):
Check the multiplication in pieces: \(0.0625(0.177978515625)=0.0111236572265625\), and \(28(0.0111236572265625)=0.31146240234375\). A binomial probability calculation with \(\operatorname{binompdf}(8,0.25,2)\) also gives approximately \(0.3115\). The student’s \(0.0311\) is about one-tenth of the correct result, consistent with a decimal-place error, not a problem with the binomial assumptions.
Conclude. Under the stated model, the probability of exactly two defective items is about \(0.3115\), or 31.15%. The event would not be considered unusual by the \(0.05\) guideline from Unusual Results in a Binomial Setting. The erroneous value \(0.0311\) would fall below that guideline and could lead to the opposite conclusion. Here the model check supports the method; it is the arithmetic that needs correction.
Correct Arithmetic Cannot Repair a Faulty Assumption
The reverse problem is just as important. A formula can be evaluated correctly and still produce an answer that does not describe the real process. For instance, a binomial model relies on a fixed success probability and independent trials. If those assumptions fail, the binomial calculation is conditional on a model that may not be appropriate.
As discussed in Independence Assumptions in Real Settings, sampling without replacement can make one outcome affect the chances on later draws. The 10% condition is one useful check when considering a binomial model for a sample drawn without replacement from a finite population. The next example makes the dependence visible by comparing the binomial calculation with a direct count of equally likely selections.
Worked Example: A Correct Binomial Calculation for the Wrong Process
A small box contains 10 cards: 3 are marked “priority,” and 7 are not. Four cards are drawn at random without replacement. A student models the number of priority cards drawn as \(X\sim B(4,0.30)\) and calculates the probability of exactly two priority cards. Is the model appropriate, and what is the probability for the actual process?
State and model check. The random variable \(X\) counts priority cards among the four selected cards. The initial chance of drawing a priority card is \(3/10=0.30\), but the draws are without replacement. If a priority card is drawn first, only 2 of the remaining 9 cards are priority; if a non-priority card is drawn first, 3 of the remaining 9 are priority. Thus the success probability changes according to earlier draws, and the draws are not independent. Also, the sample size is 4 from a population of 10, so \(4\) is greater than 10% of \(10\); the 10% condition is not satisfied. The binomial model is not justified.
Check the student’s arithmetic under the proposed model. The binomial calculation itself is:
That is the correct value for \(P(X=2)\) if four independent trials each had a 0.30 probability of success. It is not the probability for this box, because those assumptions do not match drawing four cards without replacement.
Use the actual process. Each set of four cards is equally likely. To get exactly two priority cards, choose 2 of the 3 priority cards and 2 of the 7 non-priority cards. There are \(\binom{3}{2}\binom{7}{2}=3(21)=63\) such sets. The total number of possible four-card sets is \(\binom{10}{4}=210\). Therefore:
Conclude. The probability of drawing exactly two priority cards from this box is \(0.3000\), not \(0.2646\). The student’s binomial arithmetic was accurate for the binomial model, but the model’s fixed-probability and independence assumptions were not appropriate for the actual process. The appropriate response is to use a calculation that represents the draws, not to alter the correct binomial arithmetic.
Changing Probabilities Can Matter Even Without Dependence
A fixed probability is a separate assumption from independence. Trials can be independent but have different success probabilities. In that case, the number of successes does not follow the stated binomial model with one common \(p\), even if a calculation using that model is arithmetically correct.
Averages can conceal this issue. If the success chances for two trials are 0.10 and 0.40, their average is 0.25. Replacing both chances by 0.25 creates a different model. That substitution may be convenient, but it does not generally preserve the probability of every event.
Worked Example: A Common Average Probability Hides Different Chances
A facility checks one item from each of two production shifts. The probability that an item from the morning shift is defective is 0.10; for the evening shift it is 0.40. Assume the two selected items’ defect statuses are independent. A student uses a binomial model with \(n=2\) and \(p=0.25\), the average of the two probabilities, to find the chance of exactly one defective item. Assess the model and calculate the probability for the described process.
State and model check. Let \(X\) be the number of defective items in the two-item check. Independence is given, but the defect probabilities are not the same: one is 0.10 and the other is 0.40. Therefore, the fixed success probability condition for a binomial model is not met. The average of the two probabilities does not make each trial’s probability equal to that average.
Check the binomial calculation. If both items really had independent defect probabilities of 0.25, the probability of exactly one defect would be:
The arithmetic is correct under that hypothetical binomial model. But the stated shifts have different probabilities, so this value does not represent the actual process. Calculate the two ways to have exactly one defective item: the morning item is defective and the evening item is not, or the morning item is not defective and the evening item is defective.
Conclude. For one independently selected item from each shift, the probability of exactly one defective item is \(0.42\). The binomial calculation \(0.375\) is accurate for its assumed model, but that model replaces two different chances with one common chance and therefore does not fit the process. This example shows why checking independence alone is not enough.
How to Explain Which Error Occurred
A strong explanation names the source of the problem and connects it to the result. “The answer is wrong” is not enough. State whether the process contradicts an assumption, whether the event was translated incorrectly, or whether the numerical expression was evaluated incorrectly. Then say what should be done next.
When a question gives only a model and asks for a probability under that model, calculate conditionally: “According to this model, the probability is …” If the question also asks whether the model is appropriate, evaluate the assumptions separately. A computed answer does not itself prove that those assumptions are reasonable.
Likewise, do not call every mismatch between a prediction and an observed result an arithmetic error. As explained in Using Data to Check a Probability Model, observed outcomes can differ from model expectations. A calculation error concerns the numerical work; a model concern concerns whether the probability description is suitable. Evidence of a poor fit may prompt a model review, but it is not proof that a particular arithmetic step was wrong.
Common Mistakes
- Checking only the calculator output. A calculator verifies an entered expression, not the assumptions behind it. Explain why the model fits before treating its result as a probability for the real process.
- Calling a model problem an arithmetic error. In the card example, \(0.2646\) was calculated correctly under the binomial model. The problem was sampling without replacement, which made the trials dependent and violated the 10% condition.
- Assuming independence guarantees a binomial model. The shift example has independent outcomes but different success probabilities. A binomial model also requires the same \(p\) for each trial.
- Rounding too early or shifting a decimal place. Keep sufficient digits during intermediate steps, then round the final probability consistently. Check whether the answer’s size makes sense; the first example’s intermediate product is about \(0.0111\), which must be multiplied by 28.
- Correcting the calculation when the model is the issue. Recomputing a binomial probability will not make a changing-probability process binomial. Revise the method to match the chance process.
- Overstating what a conditional answer means. Say “under the stated model” when the model has not been established as appropriate. Do not present a model-based probability as a guaranteed description of reality.
Key Takeaway
A probability conclusion needs both a suitable model and a correct calculation. Check the chance process and the model’s assumptions first; then verify the event, formula, arithmetic, and interpretation. If arithmetic is the only problem, correct the number. If an assumption fails, use a model or method that represents the actual process.
Check Your Understanding
For each situation, distinguish what should be checked about the model from what should be checked about the calculation.
- A student correctly enters \(\operatorname{binompdf}(12,0.20,3)\), but the 12 outcomes come from selecting items without replacement from a population of 60. Which model condition should be examined, and why?
- A binomial model is appropriate, but a student reports \(0.0068\) after moving the decimal in a calculation whose verified value is \(0.0680\). Is this a model-validity problem or a calculation problem? How could it affect an interpretation using the \(0.05\) guideline?
- Two independent trials have success probabilities 0.15 and 0.55. Does independence alone justify using a binomial model with \(n=2\)? Explain.
- In the card example, why is \(0.2646\) not an arithmetic error, even though it is not the probability for the actual four-card draw?
- Write one sentence that appropriately qualifies a probability calculated from a model whose assumptions have not yet been checked.